Presenting Data: Charts & Histograms
Raw data goes into a frequency table (use tally marks), then into a chart chosen to suit the data:
Pie chart — shares of a whole. Each category’s angle \(= \dfrac{\text{frequency}}{\text{total}}\times360^\circ\).
Bar chart — discrete categories, bars with gaps, heights = frequency.
Frequency polygon — plot frequency against the midpoint of each class, join with straight lines.
Histogram (equal class widths) — like a bar chart for grouped continuous data, no gaps, height = frequency.
With unequal classes, the area of each bar shows the frequency, so the height is frequency density:
\(\text{frequency density} = \dfrac{\text{frequency}}{\text{class width}}\)
Reading back: frequency = density × width.
Be ready to say why a representation suits the data (pie = proportions visible; histogram = shape of a continuous distribution) and its disadvantages (pie hides actual counts).
60 learners chose their favourite subject: Maths 18, Setswana 15, Science 12, Other 15. Find each angle.
Step 1 — one learner \(= \dfrac{360}{60} = 6^\circ\).
Step 2 — Maths \(18\times6 = 108^\circ\); Setswana \(15\times6 = 90^\circ\); Science \(12\times6 = 72^\circ\); Other \(90^\circ\).
Check: \(108+90+72+90 = 360^\circ\) ✓
Masses of 40 pumpkins: 0–2 kg (8), 2–4 kg (14), 4–6 kg (12), 6–8 kg (6). Where do the points go?
Plot frequency at the midpoints: (1, 8), (3, 14), (5, 12), (7, 6) — then join with straight ruler lines.
Times (minutes) to finish a race: 0–10 (20 runners), 10–15 (18), 15–20 (12), 20–40 (10). Find the histogram heights.
| Class | Width | Frequency | Density |
|---|---|---|---|
| 0–10 | 10 | 20 | 2.0 |
| 10–15 | 5 | 18 | 3.6 |
| 15–20 | 5 | 12 | 2.4 |
| 20–40 | 20 | 10 | 0.5 |
The 10–15 class is the most crowded (highest density) even though 0–10 has more runners — that is exactly what histograms reveal.
A pie chart shows how Kagiso spends 24 hours: sleep 8 h, school 7 h, chores 3 h, free time 6 h. Find the angle for each.
Show solution
1 hour \(= \dfrac{360}{24} = 15^\circ\). Sleep 120°, school 105°, chores 45°, free time 90°. (Sum 360° ✓)
In a histogram, the class 20 ≤ x < 35 has frequency density 1.2. (a) Find its frequency. (b) Another class, 35 ≤ x < 40, contains 14 values. Find its frequency density.
Show solution
(a) width 15: \(1.2\times15 = \mathbf{18}\).
(b) \(\dfrac{14}{5} = \mathbf{2.8}\).
Give one advantage and one disadvantage of a pie chart compared with a bar chart.
Show solution
Advantage: shows each category’s share of the whole at a glance.
Disadvantage: actual frequencies are hidden (and hard to compare across two pie charts of different totals).
- Pie angles that don’t total 360° — always check the sum.
- Frequency polygon plotted at class ends instead of midpoints.
- 🟠 Unequal-width histograms drawn with height = frequency — the tall wide bars lie; use density.
- Bar charts for continuous data with gaps between bars (or histograms with gaps).
- Chart reading and drawing are reliable Core marks; angles for pie charts almost every year.
- 🟠 Frequency density questions (complete the histogram / complete the table) are a Paper 3 standard — the table method above earns full marks.
In a pie chart of the crops grown by 90 farmers, the angle for millet is 84°. (a) How many farmers grow millet? (b) 40 farmers grow sorghum. What angle represents sorghum?
Show full solution
(a) Step 1 — the fraction of the circle: \(\dfrac{84}{360}\).
Step 2 — farmers: \(\dfrac{84}{360}\times90 = \dfrac{84}{4} = \mathbf{21}\) farmers.
(b) Step 3 — angle per farmer: \(\dfrac{360}{90} = 4^\circ\).
Step 4 — sorghum: \(40\times4 = \mathbf{160^\circ}\).
The histogram shows race-finishing times. Class 0–20 min has frequency 30. (a) Verify the frequency density shown for 0–20. (b) Find the frequency of the 20–30 class. (c) The 30–60 class contains 24 runners. Find its frequency density.
Show full solution
(a) Step 1 — density \(= \dfrac{\text{frequency}}{\text{width}} = \dfrac{30}{20} = 1.5\) ✓ matches the bar.
(b) Step 2 — frequency \(=\) density × width \(= 2.6\times10 = \mathbf{26}\) runners.
(c) Step 3 — density \(= \dfrac{24}{30} = \mathbf{0.8}\) — the “?” bar.
📓 Write your answers in your exercise book. No peeking — reveal the answers only when finished, and mark your own work. Levels: 🟢 warm-up · 🔵 exam level · 🟠 stretch.
- 🟢 A pie chart shows 72 people. What angle represents one person?
- 🟢 What angle represents a group of 18 of those 72 people?
- 🟢 A frequency polygon is plotted at the … of each class?
- 🟢 Which chart best shows shares of a whole?
- 🔵 An angle of 45° in a pie chart of 120 people represents how many people?
- 🔵 Histograms have no gaps between bars because the data is … ?
- 🟠 A class has frequency 36 and width 8. Find its frequency density.
- 🟠 A class has frequency density 2.4 and width 5. Find its frequency.
Show answers — mark your work
- \(\dfrac{360}{72} = 5^\circ\)
- \(18\times5 = 90^\circ\)
- midpoint
- A pie chart
- \(\dfrac{45}{360}\times120 = 15\)
- continuous
- \(\dfrac{36}{8} = 4.5\)
- \(2.4\times5 = 12\)